(a) How many ordered pairs (x,y) of integers are there such that \sqrt{x^2 + y^2} = 5? Does the question have a geometric interpretation?

(b) How many ordered triples (x,y,z) of integers are there such that \sqrt{x^2 + y^2 + z^2} = 7? Does the question have a geometric interpretation?

I know these questions have been asked before but the explinations are a little hard to follow and don't have a clear answer. Any help would be great!

EDIT: Okay I got the number of pairs but I don't know how to turn it into a geometric interpretation. I got a: 12 and b: 54 btw

Guest Jun 26, 2019

edited by
Guest
Jun 26, 2019

#1**+4 **

(a)

\(\sqrt{x^2+y^2}\ =\ 5\)

The solutions to this equation are all points with a distance of 5 from the origin.

So this is the equation of a circle with a radius of 5 centered at the origin.

I used a little help from here: https://math.stackexchange.com/questions/518856/integral-points-on-a-circle

There is a Pythaogrean triple with a hypotenuse of 5

A triangle with side lengths 3, 4 and 5 is a Pythagorean triple. So...

From the origin, we can go over 3 units and either up or down 4 units to reach an integer solution.

From the origin, we can go over 4 units and either up or down 3 units to reach an integer solution.

From the origin, we can go over 0 units and either up or down 5 units to reach an integer solution.

From the origin, we can go over 5 units and either up or down 0 units to reach an integer solution.

Here's a graph showing all integer solutions: https://www.desmos.com/calculator/txwzj5nmt4

There are 12 integer solutions.

hectictar Jun 26, 2019

#3**+2 **

(b) How many ordered triples (x,y,z) of integers are there such that \sqrt{x^2 + y^2 + z^2} = 7? Does the question have a geometric interpretation?

I get the following triples

6, 3, 2

But each of these can take on pos/ neg values for each integer so there are 2^3 = 8 possibilities and each of these can be arranged in 3! ways = 6 ways....so 8 * 6 = 48 possibilities

And

7 0 0 the "7" can be pos/negative = 2 possibilities and can be in any 3 positions = 2 * 3 = 6 possibilities

So.....(If I haven't missed any) we have 48 + 6 = 54 points [ just as you found ]

This is a sphere centered at (0, 0, 0) with a radius of 7

Here's a (not so good) image :

CPhill Jun 27, 2019